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Tangent and Asymptote Perpendicularity Demo

Tangent and Asymptote Perpendicularity Demo
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2015-08-01 10:28
AngleFunctionCoordinate GeometryGeometric Construction
In collections Geometric Properties of Conics
This example illustrates the geometric property that the tangent between the point of contact and the directrix subtends a right angle at the corresponding vertex in hyperbolas. It involves core math concepts like hyperbola definition, tangent properties, asymptote characteristics, and right triangle criteria. Construction highlights the use of GeoGebra commands/objects such as Point, Mirror, Hyperbola, Intersect, Tangent, Segment, and Angle. Suitable for high school analytic geometry teaching to visually understand hyperbola geometric properties and applications.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → S
  3. Mirror(S, yAxis) → S'
  4. Hyperbola(S', S, A) → c
  5. Intersect(c, yAxis, 2) → B
  6. Point(c) → P
  7. Tangent(P, c) → g, b
  8. Intersect(g, d) → D
  9. Intersect(g, f) → D'
  10. Segment(P, S) → h
  11. Segment(S, D) → i
  12. Angle(P, S, D) → α
  13. Segment(D', P) → l
  14. Segment(P, D) → m
  15. Segment(P, S') → j
  16. Segment(D', S') → k
  17. Angle(P, S', D') → β

AI Drawing Prompt Reference

Draw a hyperbola with foci at the origin and the directrix on the x-axis. Construct the vertices, foci, and directrix. Draw a point on the hyperbola and its tangent line. The tangent intersects the directrix at one point and the hyperbola again at another point. Connect these points to form line segments and calculate the angles between them. Ensure the diagram illustrates the geometric properties of the hyperbola, such as the angle between the tangent and the directrix at the vertex.
The prompt above was compiled from the original drawing commands and is for reference only.